The above animation shows a convectively unstable wave eminating from an oscillating point source. To 3-D print the final frame, go here.
Spatio-Temporal Hydrodynamic Stability Classification
For any fluid flow that can be written in terms of a dispersion relation, disturbances to the flow may be classified as either convective or absolute. Convective instabilities grow either completely downstream or upstream, leaving behind a non-growing wave. Absolute instabilities grow outward from the origin of the disturbance, eventually contaminating all regions of the domain. The distinction between convective and absolute instability is important to manufacturing process such as coating (where convective instability is desirable) and atomization of sprays (where absolute instability is desirable). The prediction of such instabilities is determined by examining the dispersion relation in the complex wave-number plane and identifying saddle points, whose associated growth-rates determine how fast the disturbance is growing and whether it is absolute or convective.
“Algorithm For Spatio-Temporal Analysis Of The Signalling Problem”. Ima Journal Of Applied Mathematics 82 (1): 1-32.
“Spatial–Temporal Stability Analysis Of Faceted Growth With Application To Horizontal Ribbon Growth”. Journal Of Crystal Growth 454.
Long-Time Algebraic Instability
Our work is motivated by the need to characterize the stability of systems that are important to the sustainable operation of processes that have exacting tolerances. The aim is to develop a simple method to predict long-time algebraic growth in linear systems, a type of instability that has been largely unexamined in the prior literature. To accomplish this task, the Fourier-Laplace integral solutions of disparate classes of partial differential equations (PDEs) are examined for structural commonality via nonstandard long-time asymptotic methods.
“Stability Of Algebraically Unstable Dispersive Flows”. Phys. Rev. Fluids 1 (7).
“Transience To Instability In A Liquid Sheet”. J. Fluid. Mech. 666: 358-390.